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Topic: Closed and exact differential forms

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	Differential geometry and topology - Wikipedia, the free encyclopedia
		Differential geometry is the study of geometry using calculus.
		Initially and up to the middle of the nineteenth century, differential geometry was studied from the extrinsic point of view: curves, surfaces were considered as lying in a Euclidean space of higher dimension (for example a surface in an ambient space of three dimensions).
		A symplectic manifold is a differentiable manifold equipped with a symplectic form (that is, a closed non-degenerate 2-form).
	en.wikipedia.org /wiki/Differential_geometry (1106 words)

	Closed and exact differential forms Info - Bored Net - Boredom (Site not responding. Last check: 2007-11-06)
		In mathematics, both in vector calculus and in differential topology, the concepts of closed form and exact form are defined for differential forms, by the equations
		= 0, to be exact is a sufficient condition to be closed.
		The implication from 'exact' to 'closed' is then a consequence of the symmetry of second derivatives, with respect to x and y.
	www.borednet.com /e/n/encyclopedia/c/cl/closed_and_exact_differential_forms.html (316 words)

	Thermodynamics and Differential Forms
		The notion of exactness is the same in both formalisms: an exact differential corresponds to an exact one-form, and an inexact differential corresponds to an inexact one-form.
		Forms that are closed, including figure 5 and figure 8, have the property that the “contour” lines in one region mesh nicely with the lines in adjacent regions.
		In this case the one-form is f and the pointy vector is ∂C/∂θ.
	www.av8n.com /physics/thermo-forms.htm (3484 words)

	Guide to math needed to study physics (Site not responding. Last check: 2007-11-06)
		Differential geometry begins with the study of differentiable manifolds, coordinate systems, vectors and tensors.
		The mathematics of differential forms, developed by Elie Cartan at the beginning of the 20th century, has been powerful technology for understanding Hamiltonian dynamics, relativity and gauge field theory.
		Cohomology is the study of the relationship between closed and exact differential forms defined on some manifold M. Students explore the generalization of Stokes' theorem, de Rham cohomology, the de Rahm complex, de Rahm's theorem and cohomology groups.
	superstringtheory.com /math/math2.html (724 words)

	De Rham cohomology - Encyclopedia, History, Geography and Biography
		It is a cohomology theory based on the existence of differential forms with prescribed properties.
		In differential geometry terminology, forms which are exterior derivatives are called exact and forms whose exterior derivatives are 0 are called closed (see closed and exact differential forms); the relationship d
		This follows by noting that exact and co-exact forms are orthogonal; the orthogonal complement then consists of forms that are both closed and co-closed: that is, of harmonic forms.
	www.arikah.net /encyclopedia/De_Rham%27s_theorem (1081 words)

	Knowledge King - De Rham cohomology (Site not responding. Last check: 2007-11-06)
		In differential geometry, differential forms on a smooth manifold which are exterior derivatives are called exact; and forms whose exterior derivatives are 0 are called closed (see closed and exact differential forms).
		Exact forms are closed, so the vector spaces of k-forms along with the exterior derivative are a cochain complex.
		The vector spaces of closed forms modulo exact forms are called the de Rham cohomology groups.
	www.knowledgeking.net /encyclopedia/d/de/de_rham_cohomology.html (154 words)

	Symplectic topology - Wikipedia, the free encyclopedia
		Symplectic topology (also called symplectic geometry) is a branch of differential topology/geometry which studies symplectic manifolds; that is, differentiable manifolds equipped with closed, nondegenerate, 2-forms.
		Symplectic topology has a number of similarities and differences with Riemannian geometry, which is the study of differentiable manifolds equipped with nondegenerate, symmetric 2-tensors (called metric tensors).
		Mikhail Gromov made, however, the important observation that symplectic manifolds do admit an abundance of compatible almost complex structures, so that they satisfy all the axioms for a complex manifold except the requirement that the transition functions be holomorphic.
	www.wikipedia.org /wiki/Symplectic_topology (317 words)

	Exact Method Marketing (Site not responding. Last check: 2007-11-06)
		Exact sequence 2: y theory, as well as in group theory, an '''exact sequence''' is a (finite or infinite) sequence] 11: The sequence is ''exact '' at A_i if the 2.
		Closed and exact differential forms 4: ology, the concepts of '''closed form''' and '''exact form''' are defined for differential form s, b 14: for an exact form, with ''andalpha;'' given and ''andbeta;'' unkno 16: It makes no real sense to ask whether a 0-form is exact, since ''d'' increases degree by 1.
		Exact functor 3: of a short exact sequence into a three-term exact sequence 4: * '''left-exact ''' if it transforms kernel (category theory)ke 5: * '''right-exact ''' if it transforms cokernel s into cokernels 6: t, i.e.
	www.elusiveeye.com /side37043-exact-method-marketing.html (689 words)

	Exterior derivative - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-06)
		In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree.
		The exterior derivative of a differential form of degree k is a differential form of degree k + 1.
		The kernel of d consists of the closed forms, and the image of the exact forms (cf.
	xahlee.org /_p/wiki/Exterior_derivative.html (287 words)

	Closed and exact differential forms -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-06)
		In abstract terms, the main interest of this pair of definitions is that asking whether this is also a necessary condition is a way of detecting (Click link for more info and facts about topological) topological information, by differential conditions.
		The implication from 'exact' to 'closed' is then a consequence of the (Click link for more info and facts about symmetry of second derivatives) symmetry of second derivatives, with respect to x and y.
		This is not true for an open ((Fungi) remnant of the partial veil that in mature mushrooms surrounds the lower part of the stem) annulus in the plane, for some 1-forms α that fail to extend smoothly to the whole disk; so that some topological condition is necessary.
	www.absoluteastronomy.com /encyclopedia/C/Cl/Closed_and_exact_differential_forms.htm (421 words)

	PlanetMath: closed differential forms on a simply connected domain
		"closed differential forms on a simply connected domain" is owned by paolini.
		Cross-references: manifolds, Poincaré lemma, continuous, curves, homotopic, regular, lemmas, exact differential form, closed differential form, simply connected, differential form, open set
		This is version 11 of closed differential forms on a simply connected domain, born on 2003-04-04, modified 2004-06-17.
	planetmath.org /encyclopedia/ClosedDifferentialFormsOnASimpleConnectedDomain.html (118 words)

	Visualizing Non-Conservative Fields and Non-Exact One-Forms
		A one-form that is the derivative of some potential is called exact and the corresponding force-field is called conservative.
		A one-form that is not exact is called non-exact or inexact.
		That is, if you go clockwise around a closed loop anywhere in the betatron electric field (figure 2), the net number of downward steps that you take is proportional to the area of the loop, independent of the shape of the loop.
	www.av8n.com /physics/non-conservative.htm (1797 words)

	De Rham cohomology (Site not responding. Last check: 2007-11-06)
		Differential forms which are exterior derivatives are called exact and forms,whose exterior derivatives are 0 are called closed.
		Differential forms which are exterior derivatives are called exact and forms, whose exterior derivatives are 0 are called closed.
		Closed forms modulo exact forms are called the de Rham cohomology groups.
	www.termsdefined.net /de/de-rham-cohomology.html (324 words)

	Alexander-Spanier cohomology - Wikipedia, the free encyclopedia
		In mathematics, particularly in algebraic topology Alexander-Spanier cohomology is a cohomology theory arising from differential forms with compact support on a manifold.
		is the vector space of closed k-forms modulo that of exact k-forms.
		They also demonstrate contravariant behavior with respect to proper maps - that is, maps such that the inverse image of every compact set is compact.
	en.wikipedia.org /wiki/Alexander-Spanier_cohomology (176 words)

	De Rham cohomology (Site not responding. Last check: 2007-11-06)
		In algebraic topology, de Rham cohomology is a cohomology theory based on the existence of differential forms with certain properties.
		In the official differential geometry terminology, forms which are exterior derivatives are called exact and forms whose exterior derivatives are 0 are called closed (see closed and exact differential forms); d
		The cohomology of the de Rham complex, that is the vector spaces of closed forms modulo exact forms, are called the de Rham cohomology groups H
	www.sciencedaily.com /encyclopedia/de_rham_cohomology (347 words)

	De Rham cohomology -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-06)
		The idea of deRham cohomology is to classify the different types of closed forms on a manifold.
		One performs this classification by saying that two closed forms α and β in are cohomologous if they differ by an exact form, that is, if is exact.
		For example, on a 2- (Commonly the lowest molding at the base of a column) torus, one may envision a constant 1-form as one where all of the "hair" is combed neatly in the same direction (and all of the "hair" having the same length).
	www.absoluteastronomy.com /encyclopedia/d/de/de_Rham_cohomology.htm (999 words)

	Stokes' theorem - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-06)
		Stokes' theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus.
		Stokes theorem then shows that closed forms defined up to an exact form can be integrated over chains defined only up to a boundary.
		The general form of the Stokes theorem using differential forms is more powerful than the special cases, of course, although the latter are more accessible and are often considered more convenient by practicing scientists and engineers.
	xahlee.org /_p/wiki/Stokes'_theorem.html (317 words)

	Course Descriptions (Site not responding. Last check: 2007-11-06)
		Definite integrals and their applications, indefinite integrals, techniques of integration, indeterminate forms, improper integrals, numerical integration, and differential equations.
		Vectors in two and three dimensions, vector-valued functions and their derivatives, surfaces, and differential calculus of functions of two or more variables and applications.
		Differential equations of first order, separable equations, exact equations, and approximation to a solution; mathematical models; linear differential equations of second order, variation of parameters, series solutions, and Laplace transforms; systems of differential equations; qualitative theory of differential equations; applications; and other related topics.
	www.avila.edu /departments/MATHWEB/pages/AvilaCourses.html (1092 words)

	Differential forms. Why? - Page 2 - Physics Help and Math Help - Physics Forums
		De Rham cohomology is heavily predicated on differential forms for instance.
		An example of a differential form (covector field) on euclidean space is df, the differential of a function, because when it sees a tangent vector at a point, it spits out a number, the directional derivative of f in the direction of the tangent vector at the point.
		An example of a vector field is the differential operator ?/?x, because when it sees a function and a point, it spits out a number, the derivative of the function in the x direction at the point.
	www.physicsforums.com /showthread.php?p=282224 (725 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		As we all know, since $d sup 2 =0$, every exact form is closed, and thus we have a quick way of generating many (in fact an infinite number of) "nice" identities.
		Of course every exact form is automatically closed, and the quotient space of closed modulo exact, for a given manifold, is the celebrated \fIde Rham cohomology\fR.
		The $c(n,k)$ for $ zeta (2)$, (13.4), is the potential function of the WZ form $omega sub {zeta (2)}$ given in (12.6), and the $c(n,k)$ for $zeta (3)$, (13.5), is the potential function for the WZ form $omega sub {zeta (3)}$ given in (12.8).
	www.math.temple.edu /~zeilberg/TROFF/pun.troff (4854 words)

	Betti number - Encyclopedia, History, Geography and Biography
		In geometric situations, the importance of the Betti numbers may arise from a different direction, namely that they predict the dimensions of vector spaces of closed differential forms modulo exact differential forms.
		The connection with the definition given above is via three basic results, de Rham's theorem and Poincaré duality (when those apply), and the universal coefficient theorem of homology theory.
		There is an alternate reading, namely that the Betti numbers give the dimensions of spaces of harmonic forms.
	www.arikah.net /encyclopedia/Betti_number (576 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Another use of closed and exact: the fact that d^2=0 is another way of expressing the fact that (mixed) partial derivatives commute.
		But it's not exact on that domain, essentially because the function it would be the differential of has to differ from @ by an additive constant on the plane minus the polar axis (y=0,x>0).
		The closed but non-exact forms then may be integrated to study the (co)homology of the domain, a type of topological classification.
	www.math.niu.edu /~rusin/known-math/99/wedge (652 words)

	Cohomological field theories (Site not responding. Last check: 2007-11-06)
		Suppose we can find a closed but not exact n-form w on N. Then we get a field theory where the Lagrangian is the pullback of w by the function f.
		L = f^(w) Since w is closed*, the integral of L over spacetime does not change when we vary f slightly, so the Euler-Lagrange equations for this field theory are trivial: every field solves the field equations.
		Charles Torre is basically considering the special case where N is an n-dimensional manifold and w is a volume form on this manifold.
	www.lns.cornell.edu /spr/2000-07/msg0026470.html (422 words)

	Math 423, Fall, 2002 (Site not responding. Last check: 2007-11-06)
		For that reason alone, both the spaces of closed forms, modulo the further subspace of exact forms, and the space of cycles modulo boundaries, have perhaps some particular significance.
		For example, the closed form dx in the torus (thought of as the square with edges identified), which is not exact, ``picks out'' the closed curve that loops around once in the horizontal direction.
		The main result of this chapter is the DeRham theorem, which states that the pairing between (real singular) homology of cycles modulo boundaries, and the cohomology of closed modulo exact forms (called deRham cohomology) is indeed an isomorphism.
	www.lehigh.edu /~dlj0/courses/423f02-lect20.html (2384 words)

	Differential Geometry, Volume 1 (Site not responding. Last check: 2007-11-06)
		The first premise is that it is absurdly inefficient to eschew the modern language of manifolds, bundles, forms, etc., which was developed precisely in order to rigorize the concepts of classical differential geometry.
		The second premise for these notes is that in order for an introduction to differential geometry to expose the geometric aspect of the subject, an historical approach is necessary; there is no point in introducing the curvature tensor without explaining how it was invented and what it has to do with curvature.
		The author has pulled together the main body of "classical differential geometry" that forms the background and origins of the state of the theory today.
	www.mathpop.com /bookhtms/dg1.htm (656 words)

	EconPapers: The Exact Cumulative Distribution Function of a Ratio of Quadratic Forms in Normal Variables with ...
		In other cases the exact CDF of a statistic of interest is very complicated despite the statistic being “simple” (for example the circular serial correlation coefficient, or a quadratic form of a vector uniformly distributed over the unit n-sphere).
		Differential geometric considerations show that there can be points where the CDF of a given statistic is not analytic, and such points do not depend on the parameters of the model but only on the properties of the statistic itself.
		The second part of the paper derives the exact CDF of a ratio of quadratic forms in normal variables, and for the first time a closed form solution is found.
	econpapers.repec.org /paper/yoryorken/01_2F02.htm (380 words)

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